What is math?

What is Math ? Interesting article below:

https://infinityplusonemath.wordpress.com/2017/06/17/what-is-math/

• Mathematics = “that which is learned“ –(Pythagoras)

Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…

What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years after Newton & Leibniz had invented it – “Epsilon-Delta” Analysis.

Difference between Riemann Integral & Lebesgue Integral ?

Unknown German Retiree Proved The “Gaussian Correlation Inequality” Conjecture

https://www.wired.com/2017/04/elusive-math-proof-found-almost-lost

$\boxed {P (a + b) \geq P (a) \times P (b)}$

Case “=” : if (a, b) independent
Case “>” : if (a, b) dependent

Thomas Royen used only high-school math (function, derivative) in his proof in 2014. He then published it in arxiv.org website – like Perelman did with the “Poincaré Conjecture”.

The genius who rejected Fields Medal & Clay Prize: Grigori Perelman

Russian Jewish mathematician Grigori Perelman proved the Poincaré Conjecture in 7 years of solitude research in his Russian apartment  – same 7 years of solitude for Andrew Wiles (The Fermat’s Last Theorem) in the Cambridge attic house and Zhang Yitang  张益唐 (7-Million-Gap Twin Primes) in the “Subway” sandwich kitchen.

7 is the Perfect Number. 1 week has 7 days,  according to the “Book of Genesis”, God created the universe in 6 days and rested “Sabbath” on the beautiful 7th day.

People involved in his journey:

• His mother who supported his early education in Math, all the way to his international fame and adulthood — Erdös Paul also had a mother supporting her single son in the entire Mathematical life.
• Prof Hamilton who “unintentionally” gave him the inspiration of the “Ricci-flow” tool – the first key to the door of The Poincaré Conjecture.
• His closed Chinese friend the MIT Math Prof  田刚 (Tian Gang) – a PhD student of Prof S.T. Yau.
• Fields Medal which he rejected — reason being he felt Prof Hamilton deserved a share of the credit.
• Clay Prize which he rejected the U\$ 1 million — for the same reason as the Fields Medal.
• Prof  S.T. Yau (邱成桐) ‘s lawsuit against “The New Yorker” jounalists on “The Poincaré Conjecture” article regarding the purported claims by his 2 Chinese mathematician students.

Trump’s Speaking Math Formula

The lower in the score the better : Trump (4.1) beats Hillary (7.7) who beats Sanders (10.1)

Trump defied most expectation from the world to win the 2017 President of the USA. His victory over the much highly educated Ivy-league Yale lawyer-trained Hilary Clinton who speaks sophisticated English is “SIMPLE English“: seldom more than 2-syllable words.

1-syllable words mostly: eg.dead, die, point, harm,…

2-syllable words to emphasize: eg. pro-blem, ser-vice, bed-lam (疯人院), root cause, …

3-syllable words to repeat (seldom): eg. tre-men-dous

His speech is of Grade-4 level, reaching out to most lower-class blue-collar workers who can resonate with him. That is a powerful political skill of reaching to the mass. Hilary Clinton’s strength of posh English is her ‘fatal’ weakness vis-a-vis connecting to the mass.

In election time, it is common to see candidates who win the heart of voters by using the local dialects of the mass, never mind they are discouraged in schools or TV: Hokkien, Teochew, etc.

Curious Thoughts in Math & Science

1. Statistical Mechanics: $e^ {- Ht}$

Quantum Mechanics: $e^{iHt}$

2. Ramanujian:

1 +2 + 3 + …+ n = –  1/12

Note: this formula is used in Quantum Physics dealing with infinity n (although it cancels out each other in subsequent calculations)

Tau Special Function:

$\boxed {\displaystyle \sum_{n=1}^{\infty}\tau (n) x^{n} = x \{(1-x)(1-x^{2})(1-x^{3})... \}^{24}}$

3. Boolean Algebra: George Boole (1847 in 《The Mathematical Analysis of Logic》) used Symbolic variables (not numbers) for Logic, inspired by Galois (1832 in Groups & Finite Fields), Hamilton’s quaternion algebra (1843).

AND$\boxed {x.y}$

NOT$\boxed {1-x}$

XOR$\boxed {x+y-2x.y}$

Extra constraints ”  $\boxed {x^{2}=x}$

4. Solomon Golomb, Sol: “Linear Feedback Shift Register” (LFSR)  — shift left the first register, fill in the back register with XOR of certain “Taps” (eg.chosen the 1st, 6th, 7th registers)

Maximal Length = The shift register of size n will repeat every $2^{n}-1$ steps (exclude all ‘0’ sequence).

Which arrangement of “Taps” will produce the maximal length ?

Solomon applied Pure Math : represent the above sequence of  registers algebraically  by:

$\boxed {x^{7}+x^{6}+ 1}$

in reducible modulo 2 (prime in polynomial, ie can’t be factored).

=> the sequence is Maximal length

LFSR Applications: Telecommunications, 3G/4G/5G, CDMA, Wifi, computers, network, signal transmission error-correction CD/DVD, Astrology : Venus-Earth distance,  etc.